Few Guardian readers will be shedding a sympathetic tear for Philip Hammond. Most will hear of his view that public sector workers are “overpaid”, reportedly shared with cabinet colleagues last Thursday, and they will recoil in outrage. As my colleague Zoe Williams forcefully argues, the chancellor’s observation – which he sought to justify by noting that those in the public sector benefited from fuller pensions than those outside it – betrays a kind of business class myopia towards the lived experience of those getting by on too little.

Hammond tried to make amends in his BBC interview yesterday: “You can’t eat your pension, you can’t feed your kids with your pension contribution, I understand that,” he said. But the damage was done. Today’s Daily Mirror seizes on the notion that Hammond is an unfeeling and hypocritical fat cat, cosy in his two grand homes, paid for by the taxpayer, and claims he rents out a third, the one he actually owns, for a juicy £10,000 a month.

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If the chancellor’s apparent indifference to nurses dependent on food banks doesn’t appal you, perhaps his reported sexism will. On Saturday, the Sun splashed on a claim that Hammond had quipped that driving a train was now so easy, “even a woman can do it”. He has vigorously denied that one, explaining that as the father of high-achieving daughters, he would never say such a thing.

Still, you can see the pattern here. Two leaks on successive days casting Hammond as a heartless boor, a hate-figure for progressive folks especially. And yet it’s the third leak, which dominates the front page of today’s Telegraph, that should give us pause. Not least because it reveals the probable motive behind the other two.

“Philip Hammond is deliberately trying to ‘frustrate’ Brexit, cabinet colleague says”, runs the headline. It makes clear that the knives are out for the chancellor, as pro-leave cabinet ministers seek to undermine, if not oust, the man they see as the biggest obstacle in the path of a hard Brexit. “What’s really going on is the establishment, the Treasury, is trying to fuck it up,” an unnamed minister tells the paper. “This is a crucial moment. That’s why we have to keep Theresa there. Otherwise the whole thing will fall apart.”

So the “overpaid” and “even a woman” stories were, it seems, part of a coordinated “get Hammond” operation (with much speculation casting Michael Gove as the driving force). On the Today programme Chris Grayling did not do much to dispel that impression: “We’re not a group of clones,” he said.

Even before the election, there were attempts to bad-mouth the chancellor – and much gloating at the loss of face he endured when he had to U-turn on his planned tax increase on the self-employed – not least because he was then the most senior remainer around the cabinet table. Hammond now shares that distinction with the de facto deputy prime minister, Damian Green, but the Tories’ non-victory on 8 June, and the resultant shrinking in authority of Theresa May, has seen his confidence grow – and with it the threat he poses to the Brexiteers.

Now he unabashedly asserts the need for a transitional period after British exit in March 2019, to avoid the economic shock of a cliff-edge Brexit. Gone are the euphemistic references to an “implementation phase”; now Hammond does not hesitate to demand an explicit transition of a “couple of years”. Another sticking point is Hammond’s preference for British involvement in the customs union, which the hardcore Brexiteers are happy to jettison along with membership of the single market.

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Put it together, and you can see why the likes of Gove, Liam Fox and Boris Johnson regard Hammond as an irritant. He wants a sane Brexit and they want him out of the way. With the chancellor removed, they would be free to drive off the cliff at top speed. With no Spreadsheet Phil to moan about the need to prioritise jobs and the economy, they could put slashing immigration first and indulge the ideological fetish that demands the eradication of the European court of justice from every last corner of British national life.

Make no mistake, Hammond is a hard sell for progressive-minded remainers. He’s a dry Thatcherite, a man who seems to have an abacus where his conscience should be. He is an unlikely champion. But right now, he is what stands in the way of the Brexiteers’ most delusional schemes. So, to adapt the words of a very different politician from a very different context, centre-left remainers have a clear task: they should simultaneously fight Hammond on austerity as if there were no Brexit, but back Hammond on Brexit as if there were no austerity.